三至五年级Discrete Mathematics计划中
Systematic listing and counting
Finding all possibilities in an organised way (outfits, routes, arrangements, handshakes) using lists, tables and tree diagrams, and checking that nothing is missed or counted twice.
math.discrete.systematic-listing
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前置知识
4 项必需的前置知识（一直追溯到底）
你将学会
- List all combinations in small problems, such as outfits from 3 shirts and 2 pairs of trousers, using a table or tree diagram.
- Explain how an organised list guarantees that no possibility is missed or repeated.
- Discover that 3 shirts and 2 pairs of trousers give 3 × 2 outfits, and generalise.
定义
- A tree diagram shows each choice as a branch, so that each path from the start to an end is one possibility.
定理
- If a first choice can be made in m ways and a second in n ways, there are m × n combined choices.
- Three different objects can be arranged in a row in exactly 6 ways.
常见误解
- Counting the same possibility twice when order does not matter, as with handshakes.
- Stopping when no new possibility comes to mind, instead of using a system.
通向
可选发现
- Combinatorics Systematic counting grows into permutations, combinations and the binomial theorem.